Prime Factors, HCF & LCM

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1. Prime Factors & Factor Trees

Every number can be broken down into prime numbers that multiply together. This is called prime factor decomposition, and it's one of the most powerful tools in number theory.

Prime Factor: A prime number that divides exactly into a given number.

Example: The prime factors of 30 are 2, 3, and 5 because 2 × 3 × 5 = 30

📌 What is a Prime Number?

Prime Number: A number with exactly 2 factors: itself and 1.

First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

📌 Method 1: Factor Tree

A factor tree breaks a number into pairs of factors until all branches end in prime numbers.

Steps:

1️⃣ Start with the number at the top

2️⃣ Split it into any two factors (not necessarily prime)

3️⃣ Split each non-prime factor into two more factors

4️⃣ Continue until all branches end in prime numbers (circle these)

5️⃣ Collect all the circled primes at the bottom

Example: Prime Factorization of 432

432 = 2 × 216
= 2 × 2 × 108
= 2 × 2 × 2 × 54
= 2 × 2 × 2 × 2 × 27
= 2 × 2 × 2 × 2 × 3 × 9
= 2 × 2 × 2 × 2 × 3 × 3 × 3
= 2⁴ × 3³

📌 Writing the Answer Using Indices/Powers

Instead of writing 2 × 2 × 2 × 2 × 3 × 3 × 3, use powers:

432 = 2⁴ × 3³

This is much cleaner! Notice: 2 appears 4 times (so 2⁴) and 3 appears 3 times (so 3³).

🧠 Key Rules:

Prime factorization is unique – any number breaks down the same way no matter which factors you choose first
Always write prime factors in ascending order (smallest to largest)
Use indices/powers to show repeated factors
The product of the prime factors always equals the original number

Q1: Express 60 as a product of its prime factors using indices.

Q2: If 144 = 2⁴ × 3², verify this is correct by multiplying it out.

2. Identifying Square & Cube Numbers

Prime factor decomposition reveals hidden patterns. If all the powers are even, it's a perfect square. If all are multiples of 3, it's a perfect cube.

📌 Perfect Square Numbers

Square Number:

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Also in the full note

3. Finding Square Roots Using Prime Factors
4. Highest Common Factor (HCF)
5. Lowest Common Multiple (LCM)
6. Worked Example – Complete Problem
📋 Chapter Summary
🧠 What to Memorise
✅ Concepts Checklist
🎯 Exam Tips & Common Mistakes

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Prime Factors, HCF & LCM